Solving Systems of Transcendental Equations Involving the Heun Functions


The Heun functions have wide application in modern physics and are expected to succeed the hypergeometrical functions in the physical problems of the 21st century. The numerical work with those functions, however, is complicated and requires filling the gaps in the theory of the Heun functions and also, creating new algorithms able to work with them efficiently. We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the Müller algorithm. The new algorithm is particularly useful in systems featuring the Heun functions and for them, the new algorithm gives distinctly better results than Newton’s and Broyden’s methods. As an example for its application in physics, the new algorithm was used to find the quasi-normal modes (QNM) of Schwarzschild black hole described by the Regge-Wheeler equation. The numerical results obtained by our method are compared with the already published QNM frequencies and are found to coincide to a great extent with them. Also discussed are the QNM of the Kerr black hole, described by the Teukolsky Master equation.

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P. Fiziev and D. Staicova, "Solving Systems of Transcendental Equations Involving the Heun Functions," American Journal of Computational Mathematics, Vol. 2 No. 2, 2012, pp. 95-105. doi: 10.4236/ajcm.2012.22013.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, “NIST Handbook of Mathematical Functions,” Cam- bridge University Press, Cambridge, 2010.
[2] S. Y. Slavyanov and W. Lay, “Special Functions, A Unified Theory Based on Singularities,” Oxford Mathematical Monographs, Oxford, 2000.
[3] K. Heun, “Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten,” Mathematische Annalen, Vol. 33, No. 2, 1889, pp. 161-179. doi:10.1007/BF01443849
[4] A. Decarreau, M. C. Dumont-Lepage, P. Maroni, A. Robert and A. Roneaux, “Formes Canoniques des Equations Confluentes de l'equation de Heun,” Annales de la Societe Scientifique de Bruxelles, Vol. 92, 1978, pp. 53.
[5] A. Decarreau, P. Maroni and A. Robert, Heun’s Differential Equations, A. Roneaux, Eds., Oxford University Press, Oxford, 1995, p. 354.
[6] M. Hortacsu, “Heun Functions and Their Uses in Physics,” 2011.
[7] P. P. Fiziev, “Novel Relations and New Properties of Con- fluent Heun’s Functions and Their Derivatives of Arbitrary Order,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, No. 3, 2010, Article ID: 035203. doi:10.1088/1751-8113/43/3/035203
[8] D. Staicova and P. Fiziev, “The Spectrum of Electro- magnetic Jets from Kerr Black Holes and Naked Singularities in the Teukolsky Perturbation Theory,” Astrophysics and Space Science, Vol. 332, No. 2, 2011, pp. 385-401. doi:10.1007/s10509-010-0520-x
[9] P. Fiziev and D. Staicova, “Application of the Confluent Heun Functions for Finding the QNMs of Nonrotating Black Hole,” Physical Review D, Vol. 84, No. 12, 2011, Article ID: 127502. doi:10.1103/PhysRevD.84.127502
[10] D. Staicova and P. Fiziev, “New Results for Electromagnetic Quasinormal Modes of Black Holes,” 2011 arXiv:1112.0310v2
[11] S. Chandrasekhar and S. L. Detweiler, “The Quasi-Nor- mal Modes of the Schwarzschild Black Hole,” Proceedings of the Royal Society A, Vol. 344, No. 1639, 1975, pp. 441-452. doi:10.1098/rspa.1975.0112
[12] S. Detweiler, “Black Holes and Gravitational Waves. III- The Resonant Frequencies of Rotating Holes,” Astro- physical Journal, Vol. 239, 1980, pp. 292-295. doi:10.1086/158109
[13] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes,” Cambridge University Press, Cambridge, 1992.
[14] G. E. Forsythe, M. A. Malcolm and C. B. Moler, “Computer Methods for Mathematical Computations,” Prentice Hall, Upper Saddle River, 1977.
[15] C. G. Broyden, “A Class of Methods for Solving Nonlinear Simultaneous Equations Math,” Mathematics of Computation, Vol. 19, 1965, pp. 577-593.
[16] P. Fiziev and D. Staicova, “Two-Dimensional Generali- zation of the Muller Root-Finding Algorithm and Its Applications,” 2011.
[17] D. E. Müller, “A Method for Solving Algebraic Equations Using an Automatic Computer,” Mathematical Tables and Other Aids to Computation, Vol. 10, No. 5, 1956, pp. 208-215.
[18] P. P. Fiziev, “Exact Solutions of Regge-Wheeler Equation and Quasi-Normal Modes of Compact Objects,” Classical and Quantum Gravity, Vol. 23, No. 7, 2006, pp. 2447-2468. doi:10.1088/0264-9381/23/7/015
[19] N. Andersson, “A Numerically Accurate Investigation of Black-Hole Normal Modes,” Proceedings: Mathematics and Physical Sciences, Vol. 439, No. 1905, 1905, pp. 47- 58.
[20] E. Berti, V. Cardoso and C. M. Will, “On Gravitational-Wave Spectroscopy of Massive Black Holes with the Space Interferometer LISA,” Physical Review D, Vol. 73, No. 6, 2006, Article ID: 064030. doi:10.1103/PhysRevD.73.064030
[21] S. Chandrasekhar, “The Mathematical Theory of Black Holes,” Oxford University Press, Oxford, 1983.
[22] V. Ferrari and L. Gualtieri, “Quasi-Normal Modes and Gravitational Wave Astronomy,” General Relativity and Gravitation, Vol. 40, No. 5, 2008, pp. 945-970. doi:10.1007/s10714-007-0585-1
[23] R. A. Konoplya and A. Zhidenko, “Quasinormal Modes of Black Holes: From Astrophysics to String Theory,” Reviews of Modern Physics, Vol. 83, No. 3, 2011, pp. 793-836. doi:10.1103/RevModPhys.83.793
[24] P. P. Fiziev, “Classes of Exact Solutions to the Teukolsky Master Equation,” Classical and Quantum Gravity, Vol. 27, No. 13, 2010, Article ID: 135001. doi:10.1088/0264-9381/27/13/135001
[25] E. W. Leaver, “An Analytic Representation for the Quasi-Normal Modes of Kerr Black Holes,” Proceedings of the Royal Society A, Vol. 402, No. 1823, 1985, pp. 285-298. doi:10.1098/rspa.1985.0119
[26] E. Berti, V. Cardoso and A. O. Starinets, “Quasinormal Modes of Black Holes and Black Branes,” Classical and Quantum Gravity, Vol. 26, No. 16, 2000, Article ID: 163001. doi:10.1088/0264-9381/26/16/163001
[27] E. Berti, “Black Hole Quasinormal Modes: Hints of Quantum Gravity?” 2004.
[28] P. P. Fiziev, “Teukolsky-Starobinsky Identities: A Novel Derivation and Generalizations,” 2009.
[29] M. Cadoni and P. Pani, “Holography of Charged Dila-Tonic Black Branes at Finite Temperature,” Journal of High Energy Physics, 2011.
[30] P. Pani, “Applications of Perturbation Theory in Black Hole Physics,” Ph.D. Thesis, Universita' degli Studi di Cagliari, Cagliari, 2011.

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